| TATM03 | Calculus D, ECTS-points /ANALYS D/ Advancement level: B | |
| Aim: To give the student familiarity with those mathematical concepts and relations introduced in the course so that they can be applied in mathematical modelling in physics, chemistry and technology. Prerequisites: Lycée mathematics (natural sciences or technical lines).Course organization: The course is divided into an introductory course and two sub-courses. Teaching is organised in the form of lectures, problem classes and seminars.Course content: Introductory course: see TATM 06. Part A Introduction: Logical and set-theoretic concepts and symbols. Number systems. Induction. Real numbers. Sets of real numbers. Supremum and infimum. Complex numbers. Polynomials. Functions of a real variable: Limits. Sequences. Continuity. Derivatives and rules of differentiation. Monotone functions. Elementary functions. Determination of primitive functions. Properties of continuous functions. Local extremal values. Rolles theorem and Lagrange's mean-value theorem. Plane curves. Tangents and asymptotes. The Riemann integral: Definition, properties and estimation. Connection between primitive function and definite integral. Integration methods. Geometrical applications: Area, length of arcs, areas and volumes of rotation. Generalised integrals. Taylor's formula, Maclaurin expansions of elementary functions with application to calculating limits, l'Hospital's rules. Ordinary differential equations: Simple first-order equations. Linear equations of arbitrary order with constant coefficients, some special equations, including Bernoulli's and Euler's equations, direction field, orthogonal trajectories. Numerical series: Comparison theorems. Absolute convergence. Convergence criteria for series and integrals. Part B Foundations of analysis: The vector space Basic topological notions. Limits of functions from to R. Continuous functions. Theorems on continuous functions defined on compact sets. Vector-valued functions. Functions of several variables: Partial derivatives. Gradients, directional derivatives and differentials. Taylor's formula. Maxima and minima. Functional determinants. Extremal values with constraints. Implicit functions. Transformation of variables. Double and triple integrals, change of variables. Geometrical applications. Generalised multiple integrals. Series: Numerical series. Convergence and absolute convergence. Comparison theorems. generalised integrals. Criteria of convergence for series and integrals. Functional sequences and functional series. Uniform convergence. Power series. Taylor series. Fourier series. The course will give proficiency in the use of concepts and relations, for example skill in the calculation of limits of given numerical and functional sequences; differentiation and integration of certain types of functions with applications to geometrical problems; transformation of variables in derivatives and integrals; the treatment of integrals which are functions of a parameter; the solution of simple first-order ordinary differential equations as well as equations with constant coefficients. Applications are given in terms of mathematical models from physics and technology.Course literature: School textbooks in natural sciences. A collection of problems and other supplementary material published by the institute. Persson, A:Analys i en variabeel. Studentlitteratur, Lund 1981. Persson, A, Böiers, L-C:.Analys i flera variaabler. Studentlitteratur, Lund 1988. | ||
| TEN 0 | Written examination | |
| TEN 1 | Written examination | |
| TEN 2 | Written examination | |